Strategy-Proof Location of Two Public Bads in
an Interval
Abhinaba Lahiri∗
Ton Storcken∗
This version, April 2016
Abstract
We consider the decision of placing two public bads in a region, modelled
by a line segment, based on a collective decision of a committee in charge.
Committee members have single-dipped preferences determined lexicographically, by the distance to the nearer and the other public bad (lexmin
preferences). A (decision) rule takes a profile of reported preferences as
input and assigns the locations of the two public bads. All rules satisfying
strategy-proofness and Pareto optimality are characterised. These rules
pick only boundary locations.
Keywords: public bads, single-dipped preferences, strategy-proofness
JEL Classification: D71
1
Introduction
We consider the problem of locating two similar noxious facilities in a region.
For example, consider the problem of locating two windmill parks along the
coast line of a country. Here the final locations are thought of as the outcome of
some rule representing the collective decision under plausible situations of the
committee in charge of the problem. In particular, we investigate the implication
of Pareto optimality and strategy-proofness for such a rule on locating two such
facilities in a given region.
The region is modelled by a unit line segment and the plausible situations
of the committee by combinations of preferences over location pairs. The later
combinations are also known as profiles. A rule selects two points from the
interval for every reported profile. We assume that each committee member
(agent) has a single-dipped lexmin preference over all possible pairs of location.
Each agent has a unique point (his dip) on the line segment as the worst location
for any of the bads for him. Preference between two different pairs of location
is determined by the distance from his dip to the nearer public bad and, in case
of a tie, by the distance to the other public bad.
In this situation, a decision rule will take the preferences of all the agents
as input, and give a pair of locations as output. We assume that there are
finitely many agents. In this paper we define the class of all decision rules
∗ Department of Quantitative Economics, Maastricht University, The Netherlands. Email
addresses: [email protected], [email protected]
1
that simultaneously satisfy two properties. Strategy-proofness - which ensures
that for every agent, truth telling is a weakly dominant strategy, and Pareto
optimality - which says that given a decision about the locations of the bads,
improving the locations for one agent would result in worsening them for other
agents.
Under these properties, we show that either bads cannot be located at an
interior point of the interval. We characterise all rules satisfying the two mentioned conditions by a pair of monotone voting rules between the boundary
points of the region. The voting rules are not independent of each other. This
class contains a variety of rules ranging from dictatorial voting to voting by
majorities. Lastly we weaken Pareto optimality to unanimity and provide an
example of a rule that selects inner points for the location of both the bads.
Our results can be seen as positive results compared to the seminal impossibility theorem of Gibbard (1973) and Satterthwaite (1975) which says that if
there are three or more alternatives, then it is impossible to find a non-dictatorial
decision rule which is also strategy proof and Pareto optimal. One way out from
this impossibility result is to consider restricted preference domains. Here a restricted domain related to single-dipped preferences is considered. Peremans
and Storcken (1999) have shown the equivalence between individual and group
strategy-proofness on domains of single-dipped preferences. Manjunath (2014)
has characterised the class of all non-dictatorial, strategy-proof and Pareto optimal decision rules when preferences are single-dipped on an interval. Barberà,
Berga and Moreno (2012) have characterised the class of all non-dictatorial,
group strategy-proof and Pareto optimal decision rules when preferences are
single-dipped on a line.
But there are impossibility results in this domain as well. Öztürk et al.
(2013, 2014) have shown that there does not exist a non-dictatorial decision rule
that is strategy-proof and Pareto optimal when preferences are single-dipped on
a disk, and on some, but not all, convex polytopes in the plane. Chatterjee
et al. (2016) have extended these results to decision rules on a sphere, when
preferences are single-dipped or, equivalently in this case, single-peaked.
All these results are about strategy-proof location of one public bad. As
far as we know, the present paper is the first one to consider the location of
two public bads in a region. Lahiri et al. (2016) has considered the problem
of joint decision of placing public bads in each of two neighboring countries,
modelled by two adjacent line segments. In that paper, they considered two different specification of single-dipped preferences, myopic extension and lexmin
extension. Under myopic extensions, they characterised the class of rules satisfying strategy-proofness, country-wise Pareto optimality, non-corruptibility and
far away condition. Finally, they compare this class with the class of rules
satisfying strategy-proofness and country-wise Pareto optimality under lexmin
extensions. The rules in the present paper bear similarities to the rules in the
last papers. There is also literature adopting a mechanism design approach to
the location of public bads, that is, including monetary side payments: e.g.,
recently, Lescop (2007) and Sakai (2012), but we are not aware of results in this
area addressing more than one noxious facilities. On the other hand, the problem of locating two public goods on an interval has been considered previously.
Ehlers (2002) considered lexicographic extension of single-peaked preference over
pairs of locations where, in order to compare two pairs of locations, an agent
first compares the best locations from each pair, and if they are same then he
2
compares the worst locations. Ehlers (2002) characterises the class of rules that
satisfies Pareto-optimality and replacement-domination.
This paper is organized as follows. Section 2 introduces the model and some
preliminary results. Section 3 shows that internal locations are excluded, and
Section 4 provides the characterisation of all rules satisfying our conditions.
Section 5 concludes.
2
Model
The objective is to locate two bads in the unit interval A = [0, 1]. The set of
possible alternatives therefore equals A = {(α, β) ∈ A × A : α ≤ β}. This
location is considered to be a collective decision of a non-empty and finite set
of agents, say N, with cardinality n ∈ N.
Each agent i ∈ N has a so called lexmin preference Rz(i) over A, characterised by its dip z(i) ∈ [0, 1] as follows. For alternatives (a1 , b1 ), (a2 , b2 ) ∈
A, (a1 , b1 ) is at least as good as (a2 , b2 ) at Rz(i) , with the usual notation
(a1 , b1 )Rz(i) (a2 , b2 ), if
min{|a1 − z(i)|, |b1 − z(i)|} > min{|a2 − z(i)|, |b2 − z(i)|}, or if
min{|a1 − z(i)|, |b1 − z(i)|} = min{|a2 − z(i)|, |b2 − z(i)|} and
max{|a1 − z(i)|, |b1 − z(i)|} ≥ max{|a2 − z(i)|, |b2 − z(i)|}
As a lexmin preference is completely determined by its dip, we identify these
with their dips and denoted by z(i) instead of Rz(i) . As usual, Pz(i) denotes the
strict or asymmetric part of Rz(i) and Iz(i) denotes the symmetric part.
A preference profile z assigns to each agent i in N a lexmin preference z(i)
over A. The set of all preference profiles is denoted by R.
For a profile z and a non-empty set S ⊆ N , let zS denote the restriction
of z to S; i.e. zS = (z(i))i∈S . For i ∈ N , profile z ′ is an i-deviation of z if
′
zN \{i} = zN
\{i} . Define the restriction of a preference z(i) to a subset C of A
by z(i)|C = (C × C) ∩ z(i). Further, define the restriction of a profile z to C
component wise; i.e., z|C = (z(i)|C )i∈N . For a, b ∈ A, we denote z|{a,b} and
z(i)|{a,b} also by z|a,b and z(i)|a,b respectively. For a ∈ A and S ⊆ N , let
(aS , zN \S ) denotes the profile, say z ′ , where for all i ∈ N \S we have z ′ (i) = z(i)
and for all i ∈ S we have z ′ (i) = a.
A rule f assigns to each preference profile z an alternative
f (z) = (αf (z), β f (z)) ∈ A such that for all z ∈ R, αf (z) ≤ β f (z). For x, y ∈ R,
µ(x, y) = x+y
2 denotes the midpoint of x and y. For any profile z ∈ R, in case
there is no confusion we write µ(z) instead of µ(αf (z), β f (z)).
We consider the following properties for a rule f .
Strategy-Proofness f is strategy-proof if for any i ∈ N and any z ∈ R and
any i-deviation z ′ of z, we have f (z)Rz(i) f (z ′ ).
Remark 1. We define a rule f to be intermediate strategy-proof if for any coalition S ⊆ N and for any profile z ∈ R and any S-deviation z ′ of z such that
z(i) = z(j) for all i, j ∈ S, we have f (z)Rz(i) f (z ′ ) for all i ∈ S. Note that
strategy-proofness and intermediate strategy-proofness are equivalent in our setting.
3
Strategy-proofness says that truth-telling is a weakly dominant strategy.
Pareto optimality Rule f is Pareto optimal if for every profile z there does
not exist an a ∈ A such that aRz(i) f (z) for all i ∈ N with aPz(j) f (z) for at
least one j ∈ N .
Monotonicity f is monotone if f (z) = f (z ′ ) for all z, z ′ ∈ R such that for all
agents i ∈ N :
• z ′ (i) ≤ z(i) ≤ αf (z) or
• 0 = αf (z) ≤ z(i) ≤ z ′ (i) < µ(z) or
• µ(z) < z ′ (i) ≤ z(i) ≤ β f (z) = 1 or
• β f (z) ≤ z(i) ≤ z ′ (i).
Monotonicity is a familiar consequence in the presence of strategy-proofness: in
this case it says, roughly, that if the preference of an agent changes such that
the chosen pair becomes better when evaluated according to the new preference,
then it remains to be chosen. As an aside, it can be shown that this monotonicity condition is weaker than what Maskin Monotonicity would demand in this
framework.
Lemma 1. Let f : R → A satisfy strategy-proofness. Then f satisfies monotonicity.
Proof. It is sufficient to prove monotonicity for an i-deviation z ′ ∈ R of z ∈ R
for an agent i ∈ N , under the following two cases.
Case 1 : z ′ (i) < z(i) ≤ αf (z)
In this case, we have the following situations
αf (z) < αf (z ′ ) : This is a violation of strategy-proofness, as agent i can
manipulate from z to z ′ .
f
α (z) > αf (z ′ ) : In this case, we have αf (z ′ ) ≤ z ′ (i) − r, where r =
αf (z) − z ′ (i), otherwise i manipulates from z ′ (i) to z(i). In turn this
implies |z(i) − αf (z ′ )| > |z(i) − αf (z)|, so we must have αf (z) =
β f (z ′ ), otherwise i manipulates from z(i) to z ′ (i). Now β f (z) ≤
z ′ (i) + (z ′ (i) − αf (z ′ )), otherwise i manipulates from z ′ (i) to z(i);
and β f (z) ≥ z(i) + (z(i) − αf (z ′ )), otherwise i manipulates from z(i)
to z ′ (i). These two inequalities combined, however, contradict the
assumption that z(i) > z ′ (i).
The only remaining possibility is αf (z) = αf (z ′ ), and by strategy-proofness
this implies β f (z) = β f (z ′ ).
Case 2 : 0 = αf (z) ≤ z(i) < z ′ (i) < µ(z)
In this case, strategy-proofness for the deviation from z to z ′ implies that
either αf (z ′ ) or β f (z ′ ) is in the interval [0, 2z(i)]. Also strategy-proofness
for the deviation from z ′ to z implies that neither αf (z ′ ) nor β f (z ′ ) is in
the interval (0, 2z ′ (i)). As z(i) < z ′ (i) < µ(z), we have 2z(i) < 2z ′ (i) <
β f (z). So it follows that [0, 2z(i)]\(0, 2z ′ (i)) = {0}. So suppose that
β f (z ′ ) = 0 = αf (z). As z ′ (i) < µ(z), we have f (z)Pz′ (i) f (z ′ ), which is a
violation of strategy-proofness for the deviation from z ′ to z. So it follows
that αf (z ′ ) = 0 and by strategy-proofness this implies β f (z ′ ) = β f (z).
This concludes the proof of Lemma 1.
4
3
No internal solution
In this section, let f be a strategy-proof and Pareto optimal rule. We show that
f cannot assign an internal location in A, i.e. the alternative chosen at any
profile is a corner point of A. Formally,
Theorem 1. Let f is a strategy-proof and Pareto optimal rule. Then for any
profile z ∈ R, f (z) ∈ {(1, 1), (0, 1), (0, 0)}.
We prove this Theorem by the following three lemmas.
The first lemma shows that if one of the two bads is located at the extreme end
of A, then the other one cannot be located at an interior point of A.
Lemma 2. For a profile z, αf (z) = 0 implies β f (z) ∈ {0, 1} and β f (z) = 1
implies αf (z) ∈ {0, 1}.
Proof. Due to symmetry, it is sufficient to prove that αf (z) = 0 implies
β f (z) ∈ {0, 1}. So suppose αf (z) = 0 but to the contrary β f (z) ∈ (0, 1).
Consider coalitions S = {i ∈ N : z(i) < µ(z)} and T = {i ∈ N : z(i) ≥ β f (z)}.
Pareto optimality of f implies that both S and T are non-empty.
Let a = maxi∈S(z) z(i). In view of Lemma 1, we may assume that
z = (µ(a, µ(z))S , 1T , zN \(S∪T ) ).
Now consider three profiles z 1 , z 2 and z 3 as follows.
z
z1
z2
z3
S
µ(a, µ(z))
µ(z)
µ(a, µ(z))
µ(z)
T
1
1
µ(β f (z), 1)
µ(β f (z), 1)
N \(S ∪ T )
z(i)
z(i)
z(i)
z(i)
The first line of this table shows the profile z. The second line shows the profile
z 1 , which is an S−deviation from z, where z 1 (i) = µ(z) for all i ∈ S. Similarly,
profiles z 2 and z 3 are defined by the proceeding lines.
Consider the deviation from z to z 1 . Strategy-proofness for this deviation implies that αf (z 1 ) = 0 and β f (z 1 ) ∈ {0, β f (z)}. So f (z 1 ) ∈ {(0, 0), f (z)}. As
(0, 0)Iz1 (i) f (z) for all i ∈ S and (0, 0)Pz1 (k) f (z) for all k ∈ N \S, Pareto optimality implies that f (z 1 ) = (0, 0).
As µ(β f (z), 1) > 21 , strategy-proofness for the deviation from z 1 to z 3 implies
that f (z 3 ) = (0, 0).
On the other hand, consider the deviation from z to z 2 . Strategy-proofness
implies that f (z 2 ) ∈ {f (z), (0, 1)}. As (0, 1)Iz2 (i) f (z) for all i ∈ T and
(0, 1)Pz2 (k) f (z) for all i ∈ N \T , Pareto optimality implies that f (z 2 ) = (0, 1).
Next consider the deviation from z 2 to z 3 . As µ(a, µ(z)) < µ(z) < 21 , strategyproofness for this deviation implies that f (z 3 ) = f (z 2 ) = (0, 1), which contradicts the fact that f (z 3 ) = (0, 0).
The second lemma shows that both bads cannot be located at a common inner
point.
Lemma 3. For the profile z, if αf (z) = β f (z) = c, then c ∈ {0, 1}.
5
Proof. Suppose αf (z) = β f (z) = c but to the contrary c ∈ (0, 1). Define a
coalitions S = {i ∈ N : z(i) ≤ c}. Note that, Pareto optimality implies that
S is a non-trivial subset of N . In view of Lemma 1, we may assume that
z = (0S , 1N \S ). Now consider the three profiles z 1 , z 2 and z 3 as follows.
z
z1
z2
z3
S
0
µ(0, c)
0
µ(0, c)
N \S
1
1
µ(c, 1)
µ(c, 1)
Consider the deviation from z to z 1 . Strategy-proofness for this deviation
implies that αf (z 1 ) ∈ {0, c}. Then Lemma 2 and strategy-proofness implies
that f (z 1 ) ∈ {(0, 0), (0, 1), f (z)}. Note that (0, 0)Iz1 (i) f (z) for all i ∈ S and
(0, 0)Pz1 (k) f (z) for all k ∈ N \S. Also (µ(c, 1), µ(c, 1))Pz1 (i) (0, 1) for all i ∈ N .
So Pareto optimality implies that f (z 1 ) = (0, 0). As µ(c, 1) > 21 , strategyproofness for the deviation from z 1 to z 3 implies that f (z 3 ) = (0, 0).
Similarly, we can derive from f (z) = (c, c), that f (z 2 ) = (1, 1), and therewith
f (z 3 ) = (1, 1), which contradicts the fact that f (z3 ) = (0, 0).
The third lemma shows that the two bads cannot be both in the interior of A.
Lemma 4. For the profile z, (αf (z), β f (z)) ∈
/ (0, 1) × [αf (z), 1).
Proof. Due to Lemma 3, it is sufficient to show that
(αf (z), β f (z)) ∈
/ (0, 1) × (αf (z), 1). First we show that in such a case we may
assume that {i ∈ N : αf (z) < z(i) < β f (z)} = ∅.
Claim 1. Suppose 0 < αf (z) < β f (z) < 1 and there exists ∅ ̸= M ⊊ N such
that αf (z) < z(i) < β f (z) for all i ∈ M . Then there exists z ′ ∈ R such that
0 < αf (z ′ ) < β f (z ′ ) < 1 and {i ∈ N : αf (z ′ ) < z ′ (i) < β f (z ′ )} = ∅.
Proof of claim 1. Define coalition T = {i ∈ N : z(i) ≥ β f (z)}. Note that
Pareto optimality implies that T is a non-trivial subset of N . Using Lemma
1, we may assume that z = (0N \(M ∪T ) , zM , 1T ). Let agent i ∈ M . Suppose
z(i) ≤ 12 . Consider the i−deviation z 1 of z, where z 1 (i) = 0. Strategy-proofness
for the deviations between z and z 1 imply that αf (z 1 ) ∈ [αf (z), β f (z)].
As 0 < αf (z) < β f (z) < 1, so Lemma 2 implies that neither αf (z 1 ) ∈ {0, 1} nor
β f (z 1 ) ∈ {0, 1}. So it follows that 0 < αf (z 1 ) ≤ β f (z 1 ) < 1. Then Lemma 3
implies 0 < αf (z 1 ) < β f (z 1 ) < 1. If z(i) > 21 , then consider the i−deviation z 1
of z, where z 1 (i) = 1. By a similar argument we have 0 < αf (z 1 ) < β f (z 1 ) < 1.
Define M 1 = {i ∈ N : αf (z 1 ) < z 1 (i) < β f (z 1 )}. Now if M 1 = ∅, then this
concludes the proof of Claim 1. Otherwise repeat the procedure above with M 1
in the role of M and construct M 2 . As |M | < n, there exists a finite k ∈ N such
that M k = ∅. This concludes the proof of Claim 1.
Now suppose 0 < αf (z) < β f (z) < 1. In view of Claim 1 and Lemma 1, we can
assume that there is a coalition S, such that z = (0S , 1N \S ). Pareto optimality
implies that S is a non -trivial coalition. But now (µ(z), µ(z)) Pareto dominates
f (z). This contradiction proves Lemma 4.
Proof of Theorem 1. Follows from Lemmas 2, 3 and 4.
6
4
Characterisation
In this section, we characterise the class of rules satisfying strategy-proofness
and Pareto optimality. Note that because of Theorem 1, the range of any such
rule is a subset of B = {00, 01, 11}, where 00 denotes (0, 0) and so on. Restricted
to B, we have the following preferences.
Dips
z(i) < 0.5
z(i) = 0.5
z(i) > 0.5
Preferences
11Pz(i) 01Pz(i) 00
11Iz(i) 01Iz(i) 00
00Pz(i) 01Pz(i) 11
Note that if z(i) < 0.5, then 11 is the unique top ranked alternative of Rz(i) .
Similarly, if z(i) > 0.5, then 00 is the unique top ranked alternative of Rz(i) .
On the other hand if z(i) = 0.5, then the top ranked alternatives of Rz(i) is the
set B. This brings us to the following lemma.
Lemma 5. Let f be a strategy-proof and Pareto optimal rule. Then
f (z) = f (z ′ ) for any z, z ′ ∈ R such that z|B = z ′ |B .
Proof. Follows from Theorem 1 and strategy-proofness.
Next we introduce a strong monotonicity property of a rule f as follows.
Strong Monotonicity Rule f is strongly monotone if αf (z) ≥ αf (z ′ ) and
β f (z) ≥ β f (z ′ ) for all i ∈ N and for all profiles z, z ′ ∈ R such that
z(i) ≤ z ′ (i).
Next lemma shows an implication of strategy-proofness and Pareto optimality.
Lemma 6. Suppose f be a strategy-proof and Pareto optimal rule. Then f is
strongly monotone.
Proof. As f is strategy-proof and Pareto optimal, Theorem 1 implies that f (z) ∈
B for any z ∈ R. It is sufficient to show that αf (z) ≥ αf (z ′ ) and β f (z) ≥ β f (z ′ ),
where z ′ ∈ R is an i−deviation of z ∈ R such that z(i) < z ′ (i). Now consider
the following cases.
z(i) < 12 : In this case, we have 11Pz(i) 01Pz(i) 00. Hence strategy-proofness for
the deviation from z to z ′ implies that αf (z) ≥ αf (z ′ ) and β f (z) ≥ β f (z ′ ).
z ′ (i) > 12 : In this case, we have 00Pz′ (i) 01Pz′ (i) 11. Hence strategy-proofness
for the deviation from z ′ to z implies that αf (z) ≥ αf (z ′ ) and β f (z) ≥
β f (z ′ ).
This concludes the proof of Lemma 6.
Next lemma shows the opposite direction.
Lemma 7. Suppose f be a strongly monotone rule. Also suppose that f (z) ∈ B
for all z ∈ R. Then f is strategy-proof.
Proof. Suppose z ′ ∈ R be an i−deviation of some z ∈ R. Without loss of
generality, assume that z(i) < z ′ (i). Now we consider the following cases.
7
If z(i) = 12 , hence 11Iz(i) 01Iz(i) 00, strategy-proofness follows as i is indifferent
between all the elements in B.
If z(i) > 12 , then strategy-proofness follows because f (z) = f (z ′ ) by Lemma 5.
If z(i) < 12 , hence 11Pz(i) 01Pz(i) 00, strategy-proofness follows as by monotonicity α(z ′ ) ≤ α(z)
This concludes the proof of Lemma 7.
Now we define a class of rules as follows. Each rule is characterised by two
families of pairs of coalitions Wα , Wβ ∈ 2N × 2N . The first coalition in such
a pair can be thought of as the collection of agents for whom 11 is the unique
top ranked alternative. The second coalition in such a pair can be thought of
as the collection of agents who are indifferent among all alternatives in B. Wα
can be thought of as the collection of such pairs who are decisive to locate αf ()
at 1. Similarly Wβ can be thought of as the collection of such pairs who are
decisive to locate β f () at 1. We define two such families Wα , Wβ as decisive if
they satisfy the following properties.
Inclusion : Wα ⊆ Wβ .
This follows from the fact that αf () ≤ β f ().
Monotonicity : For (S, T ) ∈ Wα (Wβ ), we have (S ′ , T ′ ) ∈ Wα (Wβ ) whenever
S ⊆ S ′ and S ∪ T ⊆ S ′ ∪ T ′ .
This is a direct translation of the strong monotonicity property introduced
previously in this section.
Boundary condition : (X, N \X) ∈ Wα for all non-empty X ⊆ N , and (∅, Y ) ∈
/
Wβ for all Y ⊊ N .
This is an implication of Pareto optimality, restricted to the boundary
points.
Non-compromising at maximal conflict : T ̸= ∅ for all (S, T ) ∈ Wβ \Wα .
This property ensures that 01 cannot be selected if there are no agents
indifferent among all the alternatives in B. The fact that this holds for any
strategy-proof and Pareto optimal rule can be seen as follows. Suppose f
be a strategy-proof and Pareto optimal rule, but to the contrary f (z) = 01
for some z ∈ R with z(i) ̸= 0.5 for all i ∈ N . This implies that in the
profile z, there are no agents indifferent among all the alternatives in B.
Now consider another profile z ⋆ ∈ R as follows. z ⋆ (i) = 0 if z(i) < 0.5
and z ⋆ (i) = 1 if z(i) > 0.5. Note that z|B = z ⋆ |B . As f is strategyproof and Pareto optimal, Lemma 5 implies that f (z) = f (z ⋆ ) = 01. This
contradicts Pareto optimality as (α, α)Pz⋆ (i) (0, 1) for all i ∈ N and for
any α ∈ (0, 1).
Based on these two families, we define a rule f Wα Wβ as follows. For any profile
z ∈ R, define
S(z) = {i ∈ N : z(i) < 12 }.
T (z) = {i ∈ N : z(i) = 12 }.
8

00





01
f Wα Wβ (z) =





11
if (S(z), T (z)) ∈
/ Wβ
if (S(z), T (z)) ∈ Wβ \Wα
if (S(z), T (z)) ∈ Wα
This brings us to our final theorem.
Theorem 2. Let f : R −→ A be a rule. Then f is strategy-proof and Pareto
optimal if and only if there exist two families of pairs of coalitions Wα and Wβ ,
which are decisive; such that f (z) = f Wα Wβ (z) for all z ∈ R.
We prove this theorem with the help of the following two lemmas. The first
lemma shows the only if direction of Theorem 2. For any rule f , define
{
}
αf (z) = 1 and
.
Wαf = (S, T ) ∈ 2N × 2N : ∃ z ∈ R with
S = S(z) and T = T (z)
{
}
β f (z) = 1 and
Wβf = (S, T ) ∈ 2N × 2N : ∃ z ∈ R with
.
S = S(z) and T = T (z)
This brings us to the following lemma.
Lemma 8. Suppose f : R −→ A be a strategy-proof and Pareto optimal rule.
Then Wαf and Wβf are decisive.
Proof. As f is strategy-proof and Pareto optimal, Wαf and Wβf are well defined
sets. Next we show that Wαf and Wβf are decisive. Note that the inclusion
property follows from αf (z) ≤ β f (z). As f is strategy-proof and Pareto optimal, Lemma 6 implies that f is strongly monotone. This in turn implies the
monotonicity property. Also the fact that Wαf and Wβf are non-compromising
at maximal conflict and satisfies the boundary condition follows directly from
Pareto optimality of f .
The next lemma shows the if direction of Theorem 2.
Lemma 9. Suppose Wα ⊂ 2N × 2N and Wβ ⊂ 2N × 2N be two families of
pairs of coalitions which are decisive. Then f Wα Wβ is strategy-proof and Pareto
optimal.
Proof. Note that because of the inclusion property, f Wα Wβ is a well defined
function. Next we show that f Wα Wβ is strategy-proof.
Proof of strategy-proofness. Note that f Wα Wβ (z) ∈ B for all z ∈ R. Then in
view of Lemma 7, it is sufficient to show that f Wα Wβ is strongly monotone.
Consider z, z ′ ∈ R such that z(i) ≤ z ′ (i) for all i ∈ N . This implies that either
S(z) ⊆ S(z ′ ) or S(z) ∪ T (z) ⊆ S(z ′ ) ∪ T (z ′ ). Then monotonicity of Wα and Wβ
yields that f Wα Wβ is strongly monotone.
This shows that f Wα Wβ is strategy-proof. Next we show that f Wα Wβ is Pareto
optimal.
Proof of Pareto optimality. Note that f Wα Wβ is Pareto optimal if for any z ∈ R;
either there is an agent j ∈ N for whom f Wα Wβ (z) is the unique top ranked
alternative at Rz(j) ; or f Wα Wβ (z) is one of the top ranked alternatives of all
agents. As f Wα Wβ (z) ∈ B for all z ∈ R, we distinguish the following cases.
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Case I : f Wα Wβ (z) = 11
We are done if there exists an agent j ∈ N such that z(j) < 0.5. If z(i) ≥
0.5 for all i ∈ N , by the boundary condition it follows that z(i) = 0.5 for
all i ∈ N . But then 11 is among the top ranked alternatives of all agents
and the alternative is Pareto optimal at this profile.
Case II : f Wα Wβ (z) = 00
Similar to the case I.
Case III : f Wα Wβ (z) = 01
In this case, the non-compromising at maximal conflict property implies
that T (z) ̸= ∅. Note that B is the set of all top ranked alternatives for
all agents in T (z). So 01 cannot be improved by any alternatives in A\B.
Now suppose, for contradiction that 11 is weakly better than 01 for all
agents and 11 is strictly better than 01 for atleast one agent j ∈ N .
This implies that S(z) ∪ T (z) = N and S(z) ̸= ∅. Then the boundary
condition implies that f Wα Wβ (z) = 11, which contradicts our assumption
that f Wα Wβ (z) = 01. Similarly suppose, for contradiction that 00 is
weakly better than 01 for all agents and 00 is strictly better than 01 for
atleast one agent j ∈ N . This implies that S(z) = ∅ and T (z) ̸= N . Then
the boundary condition implies that f Wα Wβ (z) = 00, which contradicts
our assumption that f Wα Wβ (z) = 01. So we can conclude that 01 is a
Pareto optimal outcome at this profile.
Combining these cases yields that f Wα Wβ is Pareto optimal and concludes the
proof of Lemma 9.
Proof of Theorem 2. In view of Lemmas 8 and 9, it is sufficient to show that
f
f
f (z) = f Wα Wβ (z) for all z ∈ R.
Note that αf (z) = 1(β f (z) = 1) ⇔ (S(z), T (z)) ∈ Wαf (Wβf ). This shows that
f
f (z) = f Wα Wβ (z) for all z ∈ R and concludes the proof of Theorem 2.
f
5
Conclusion
In this section, first we provide an example of a non-dictatorial rule that belongs
to the class described in Section 4.
Example 1. Define two families of pairs of coalitions (Vα and Vβ ) as follows.
}
{
either S ∪ T = N and S ̸= ∅
.
Vα = (S, T ) ∈ 2N × 2N :
or |S| > 3n
4 and T ̸= ∅
}
{
Vβ = Vα ∪ (S, T ) ∈ 2N × 2N : |S| > n4 and T ̸= ∅ .
Note that the two families Vα and Vβ are decisive. So Theorem 2 implies that
f Vα Vβ is strategy-proof and Pareto optimal.
Theorem 2 characterises the class of rules in our model. Although, the class
contain rules other than dictatorships, this is indeed limited. These rules do
not select inner points for any profile of reported preferences. Now suppose
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we consider a weakening of Pareto optimality, namely unanimity, which says
that if at a given profile there exists atleast one alternative which is common
among the top ranked alternatives of all agents, then the rule should select one
of those alternative at that profile. Then we can show that Theorem 2 does
not hold any more. In our domain of preferences, strategy-proofness together
with unanimity does not imply Pareto optimality. We show this by means of
the following example.
Example 2. This rule is characterised by an alternative (α, β) ∈ (0, 1) × (0, 1).
For any profile z ∈ R, define
N1 (z) = {i ∈ N : z(i) ≤ 21 }.
N0 (z) = {i ∈ N : z(i) > 21 }.
yl (z) = max z(i).
i∈N1 (z)
yu (z) = min z(i).
i∈N0 (z)
h(α,β) (z) =

(1, 1)








(0, 0)







(α, β)







(0, 1)
if z(i) ≤
1
2
for all i ∈ N
if z(i) ≥ 12 for all i ∈ N
with z(j) > 12 for atleast one j ∈ N
if 2yl (z) < α ≤ β < 2yu (z) − 1
otherwise
Note that this rule is unanimous. Next, we show that this rule satisfies strategyproofness. Without loss of generality consider a profile z such that there exists an i ∈ N with z(i) < 12 . Consider an i−deviation z ′ of z and assume
that h(α,β) (z) ̸= h(α,β) (z ′ ). We have to show that h(α,β) (z)Rz(i) h(α,β) (z ′ ).
As z(i) < 21 , so (1, 1) is the unique top ranked alternative of agent i according to the preference Rz(i) . So, if h(α,β) (z) = (1, 1), then it follows that
h(α,β) (z)Pz(i) h(α,β) (z ′ ). So assume that h(α,β) (z) ̸= (1, 1). As z(i) < 21 , we
can conclude that h(α,β) (z) ̸= (0, 0). Also as h(α,β) (z) ̸= (1, 1), there exists
an agent j ∈ N \{i} such that z(j) > 12 . Without loss of generality, assume
that yl (z) = z(i) and yu (z) = z(j) As z ′ is an i−deviation of z, it follows that
h(α,β) (z ′ ) ̸= (1, 1). Now we consider the following cases.
h(α,β) (z) = (α, β) :
In this case, 2yl (z) < α ≤ β < 2yu (z)−1. This implies |α − z(i)| > |0 − z(i)|.
So we can conclude that (α, β)Pz(i) (0, 0) and (α, β)Pz(i) (0, 1). So in this
case we can conclude that h(α,β) (z)Pz(i) h(α,β) (z ′ ).
h(α,β) (z) = (0, 1) :
Note that if h(α,β) (z ′ ) = (0, 0) then we have h(α,β) (z)Pz(i) h(α,β) (z ′ ). So
we consider the following sub cases.
α ≤ 2yl (z) = 2z(i) : In this situation, as β < 1, we have (0, 1)Pz(i) (α, β).
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α > 2yl (z) = 2z(i) : In this situation, as h(α,β) (z) = (0, 1), so we have
2yl (z) − 1 = 2z(j) − 1 ≤ β. Then from the definition of h(α,β) it
follows that agent i cannot change the outcome to (α, β) by unilateral
deviation.
From these sub cases, it follows that in this case we have h(α,β) (z)Pz(i) h(α,β) (z ′ ).
Also note that if z(i) = 21 , then h(α,β) (z) ̸= (α, β), and the remaining three
alternatives are his top ranked alternatives. So we can conclude that this rule
is strategy-proof. Evidently, this rule is not Pareto optimal.
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